# Differentiable function

A function f(x) is **differentiable** at the point *a* if and only if, as *x* approaches *a* (which it is never allowed to reach), the value of the quotient:

- <math>\frac{f(x) - f(a)}{(x - a)}</math>

approaches a limiting value that we call the derivative of the function f(x) at *x=a*.

There is also the more rigorous <math>\epsilon-\delta</math> definition: a function f is said to be differntiable at point *a* if ∀<math>\epsilon>0</math> ∃<math>\delta>0</math> such that if

- <math> |x - a| < \delta\,</math>

then

- <math>|\frac{f(x) - f(a)}{x-a} - f'(a) | < \epsilon \,</math>.